Partial Differential Equations

By introducing continuous cellular automata with a continuous range of gray levels, we have successfully removed some of the discreteness that exists in ordinary cellular automata. But there is nevertheless much discreteness that remains: for a continuous cellular automaton is still made up of discrete cells that are updated in discrete time steps.

So can one in fact construct systems in which there is absolutely no such discreteness? The answer, it turns out, is that at least in principle one can, although to do so requires a somewhat higher level of mathematical abstraction than has so far been necessary in this book.

The basic idea is to imagine that a quantity such as gray level can be set up to vary continuously in space and time. And what this means is that instead of just having gray levels in discrete cells at discrete time steps, one supposes that there exists a definite gray level at absolutely every point in space and every moment in time—as if one took the limit of an infinite collection of cells and time steps, with each cell being an infinitesimal size, and each time step lasting an infinitesimal time.

But how does one give rules for the evolution of such a system? Having no explicit time steps to work with, one must instead just specify the rate at which the gray level changes with time at every point in space. And typically one gives this rate as a simple formula that depends on the gray level at each point in space, and on the rate at which that gray level changes with position.

Such rules are known in mathematics as partial differential equations, and in fact they have been widely studied for about two hundred years. Indeed, it turns out that almost all the traditional mathematical models that have been used in physics and other areas of science are ultimately based on partial differential equations. Thus, for example, Maxwell's equations for electromagnetism, Einstein's equations for gravity, Schrödinger's equation for quantum mechanics and the Hodgkin–Huxley equation for the electrochemistry of nerve cells are all examples of partial differential equations.

It is in a sense surprising that systems which involve such a high level of mathematical abstraction should have become so widely used